Monte Carlo results for random coagulation

Exact combinatorial approach to finite coagulating systems

This paper outlines an exact combinatorial approach to finite coagulating systems. In this approach, cluster sizes and time are discrete and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters, an exact expression is derived for the probability of a coagulating system with an arbitrary kernel being found in a given cluster configuration when monodisperse initial conditions are applied. Then this probability is used to calculate the time-dependent distribution for the number of clusters of a given size, the average number of such clusters, and that average's standard deviation. The correctness of our general expressions is proved based on the (analytical and numerical) results obtained for systems with the constant kernel. In addition, the results obtained are compared with the results arising from the solutions to the mean-field Smoluchowski coagulation equation, indicating its weak points. The paper closes with a brief discussion on the extensibility to other systems of the approach presented herein, emphasizing the issue of arbitrary initial conditions.

Exact kinetics of the sol-gel transition

The formation of a gel in a disperse system wherein binary coagulation alone governs the temporal changes of particle mass spectra is studied under the assumption that the coagulation kernel is proportional to the product of masses of coalescing particles. This model is known to reveal the sol-gel transition, i.e., the formation of one giant cluster with the mass comparable to the total mass of the whole system. This paper reports on the exact solution of this model for a finite total mass of the coagulating system. The evolution equation for the generating functional defining all properties of coagulating systems is solved exactly for this particular kernel. The final output is the exact expression for the single-particle mass spectrum as a function of time. The analysis of the spectrum in the thermodynamic limit shows that after a critical time a giant single particle (the gel) appears. Although the concentration of this giant gel particle is zero in the thermodynamic limit, it actively interacts with smaller particles "eating" them and thus growing in mass. Special attention is given to the transition point, where the gel is appearing. It is demonstrated that the sol-gel transition reminds the second-order phase transition. The time dependencies of the gel mass, the number concentration, and the second moment of the particle mass spectrum are found.

Finite-size fluctuations in interacting particle systems

Fluctuations may govern the fate of an interacting particle system even on the mean-field level. This is demonstrated via a three species cyclic trapping reaction with a large, yet finite number of particles, where the final number of particles N(f) scales logarithmically with the system size N, N(f) approximately ln N. Statistical fluctuations, that become significant as the number of particles diminishes, are responsible for this behavior. This phenomenon underlies a broad range of interacting particle systems including in particular multispecies annihilation processes.

Exchange-driven growth

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) approximately exp(-x(2-nu)), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x) approximately x(-5). In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T approximately [lnN](-(nu-2)), in the large system size limit N--> infinity. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.

Kinetics of random aggregation-fragmentation processes with multiple components

A computationally efficient algorithm is presented for exact simulation of the stochastic time evolution of spatially homogeneous aggregation-fragmentation processes featuring multiple components or conservation laws. The algorithm can predict the average size and composition distributions of aggregating particles as well as their fluctuations, regardless of the functional form (e.g., composition dependence) of the aggregation or fragmentation kernels. Furthermore, it accurately predicts the complete time evolutions of all moments of the size and composition distributions, even for systems that exhibit gel transitions. We demonstrate the robustness and utility of the algorithm in case studies of linear and branched polymerization processes, the last of which is a two-component process. These simulation results provide the stochastic description of these processes and give new insights into their gel transitions, fluctuations, and long-time behavior when deterministic approaches to aggregation kinetics may not be reliable.

Hybrid Monte Carlo Method for Simulation of Two-Component Aerosol Coagulation and Phase Segregation

The paper presents the development of a hybrid Monte Carlo (MC) method for the simulation of the simultaneous coagulation and phase segregation of an immiscible two-component binary aerosol. The model is intended to qualitatively model our prior studies of the synthesis of mixed metal oxides for which phase-segregated domains have been observed in molten nanodroplets. In our previous works (J. Aerosol Sci.32, 1479 (2001); Chem. Eng. Sci.56, 5763 (2001); submitted for publication) we developed sectional and monodisperse models where the internal state of the aerosol particles was described. These methods have certain limitations and it is difficult to include additional physical effects into the framework. Our new approach combines both constant volume and constant number Monte Carlo methods. Similar to our previous models, we assume that the phase segregation is kinetically controlled. The MC approach allows us to compute the mean number of enclosures (minor phase) per droplet, average enclosure volume, and the width of the enclosure size distribution. The results show that asymptotic behavior of enclosure distribution exists that is independent of initial conditions, which is very close to the continuum self-preserving distribution. Temperature is a key parameter because it allows for a significant change in the internal transport rate within each droplet. In particular, increasing the temperature significantly enhances the Brownian coagulation rate and lowers the number of enclosures per droplet. As a result, the MC results indicate that the growth of the minor phase can be moderated quite dramatically by small changes in system temperature. These results serve to illustrate the utility of this synthesis approach to the controlled growth of nanoparticles through the use of a majority matrix to slow down the encounter frequency of the minor phase and therefore its particle size.

Monte Carlo Simulation of Particle Aggregation and Simultaneous Restructuring

Ultrafine ("nano"-) particles produced from highly supersaturated vapors or liquids often undergo rapid coagulation and slow interspherule coalescence. Resulting "aggregates" typically contain hundreds of small spherules bound together in tenuous structures characterized by mass fractal dimensions much less than 3. Such aggregates have large and relatively accessible initial surface area but are metastable with respect to more compact configurations, especially in high temperature environments (e.g., flames). Subject to deliberately idealized "uncoupled" rate laws for coagulation and coalescence, we illustrate the power of Monte Carlo simulation methods to obtain the self-preserving joint distribution function (with respect to both particle size and surface area) of populations of coagulating fractal aggregates in the continuum regime, simultaneously undergoing finite-rate restructuring (e.g., via surface-energy-driven viscous flow). Unconditional distributions with respect to either particle volume or area are also obtained from the Monte Carlo simulations. These are conveniently quantified by fitting them to log-normal distributions and we report the sensitivity of the associated spreads to characteristic fusion/coagulation time ratio, chi, and particle fractal dimension, Df, here prespecified. We also calculate and report selected "mixed" moments of the joint pdf with respect to particle volume and surface area needed for engineering calculations of deposition or diffusion-controlled vapor scavenging, as well as the important ratio of actual mean area to that area corresponding to the mean particle volume in the aerosol population. This work sets the stage for tractable simulations of particle dynamics in more complex coagulating systems requiring multi-internal (state-) variables for their more realistic and self-consistent description. Copyright 1999 Academic Press.